fluids Review On the Values for the Turbulent Schmidt Number in Environmental Flows Carlo Gualtieri 1,*, Athanasios Angeloudis 2, Fabian Bombardelli 3, Sanjeev Jha 4 and Thorsten Stoesser 5 1 Department of Civil, Construction and Environmental Engineering (DICEA), University of Napoli “Federico II”, Napoli 80125, Italy 2 Department of Earth Science and Engineering, Faculty of Engineering, Imperial College, SW7 2AZ London, UK; [email protected] 3 Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA; [email protected] 4 Department of Civil Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada; [email protected] 5 Hydro-environmental Research Centre, School of Engineering, Cardiff University, The Parade, CF24 3AA Cardiff, UK; [email protected] * Correspondence: [email protected] Academic Editor: Meir Teitel Received: 25 February 2017; Accepted: 17 April 2017; Published: 19 April 2017 Abstract: Computational Fluid Dynamics (CFD) has consolidated as a tool to provide understanding and quantitative information regarding many complex environmental flows. The accuracy and reliability of CFD modelling results oftentimes come under scrutiny because of issues in the implementation of and input data for those simulations. Regarding the input data, if an approach based on the Reynolds-Averaged Navier-Stokes (RANS) equations is applied, the turbulent scalar fluxes are generally estimated by assuming the standard gradient diffusion hypothesis (SGDH), which requires the definition of the turbulent Schmidt number, Sct (the ratio of momentum diffusivity to mass diffusivity in the turbulent flow). However, no universally-accepted values of this parameter have been established or, more importantly, methodologies for its computation have been provided. This paper firstly presents a review of previous studies about Sct in environmental flows, involving both water and air systems. Secondly, three case studies are presented where the key role of a correct parameterization of the turbulent Schmidt number is pointed out. These include: (1) transverse mixing in a shallow water flow; (2) tracer transport in a contact tank; and (3) sediment transport in suspension. An overall picture on the use of the Schmidt number in CFD emerges from the paper. Keywords: environmental fluid mechanics; computational fluid dynamics; Reynolds-averaged Navier-Stokes equations (RANS); turbulent Schmidt number 1. Introduction Environmental Fluid Mechanics (EFM) is the scientific branch that studies naturally-occurring fluid flows of air and water on our planet Earth, especially those that affect the environmental quality of air and water [1]. Scales of relevance within EFM range from millimeters to kilometers and from seconds to years [1]. EFM is generally not aimed at design, but at understanding and prediction of flow features, which in turn may be useful within a decision-making context. Indeed, typical problems in EFM concern the prediction of parameters of environmental quality that depend on natural fluid flows, such as sediment transport in rivers and pollution levels in an airshed, for example. Research methods in EFM include: (a) full-scale on-site experiments; (b) full-scale laboratory Fluids 2017, 2, 17; doi:10.3390/fluids2020017 www.mdpi.com/journal/fluids Fluids 2017, 2, 17 2 of 27 experiments; (c) reduced-scale laboratory experiments in wind tunnels and water flumes; as well as (d) analytical, semi-empirical modelling and computer simulations with Computational Fluid Dynamics (CFD). The main advantages of CFD are that it: (1) allows full control over the boundary conditions; (2) provides results in every point of the computational domain at all times (“whole-flow field data”); and (3) does not suffer from potentially-incompatible similitude requirements due to scaling limitations since simulations can be performed at full scale. CFD also allows efficient parametric analysis of different configurations and for different conditions. However, it is not that CFD is free from potential issues. The accuracy and reliability of CFD solutions are oftentimes questioned, and thus, solid verification and validation studies are necessary [2–4]. In this regard, high-quality experimental data for validation studies and analytical solutions become indispensable. The uncertainty in any modeling activity can be determined as follows [5]. d = dmodel + dnumerical + dinput − dmeasurements (1) In other words, the total uncertainty in any simulation (d) is the sum of the model structural uncertainty (dmodel); the uncertainty induced by numerical aspects of solving the equations of the model (dnumerical); the uncertainty in the initial/boundary conditions and parameters (dinput); minus the uncertainty due to the measurements (dmeasurements). In the arena of EFM, some criteria for the application of the verification and validation techniques have been proposed by [5–10]. More recently, [2] suggested applying the guidelines for model development and evaluation first addressed by [3], i.e., the so-called ten-step approach to CFD methods for EFM flows. This is of particular importance if turbulent flow with mass transfer is modelled, as is typical in the field of EFM. The most widely-applied approach for simulating turbulent flows is that based on the Reynolds-Averaged Navier-Stokes (RANS) equations, where the turbulent scalar fluxes are mostly estimated by assuming the Standard Gradient Diffusion Hypothesis (SGDH) [11,12]. ¶C u0 C0 = −Dt−x ¶x ¶C v0 C0 = −Dt−y ¶y (2) ¶C w0 C0 = −Dt−z ¶z where u, v and w are velocity components in the x, y and z directions, respectively, and C is the solute concentration. The overbar indicates time-averaged quantities, while the prime stands for fluctuating quantities. Finally, in Equation (2), Dt-x, Dt-y and Dt-z are the turbulent diffusivities in the x, y and z directions, respectively, of the scalar being transported within the turbulent flow. This approach is a natural extension of the Fickian formulation for laminar fluxes; thus, the same name is used in the context of turbulent flows. The SGDH relates the scalar Reynolds fluxes to the spatial gradient of the time-averaged concentration through the turbulent diffusivity. The SGDH model adopts a scalar, the eddy-diffusivity, assuming an alignment of the scalar flux vector with the mean scalar gradient; in Equation (2), the possibility of having different diffusivities in each direction is explored with the use of a diffusivity tensor, which is both diagonal and anisotropic. However, in many cases, this assumption is not verified. Alternatively, other tensorial eddy-diffusivity approaches could be introduced to achieve more accurate predictions of scalar transport and dispersion in complex flows. Such a type of model is given by the so-called Generalized-Gradient-Diffusion Hypothesis (GGDH) [13] and also the High-Order extension of the GGDH model (HO-GGDH) [14]. More generally, it has been recognized for a long time now that the width of a dispersing patch in a turbulent environment grows proportionally with time, rather than the square root of time, if not a little faster [15]. Recently, a new approach using a non-local quantity, which may be interpreted in terms of the probability density distribution of the turbulent velocity, has been proposed [16]. This approach assumes that the diffusivity grows with the patch size and that eddies of the scale most comparable to the size of the patch are the most effective at distorting and dispersing it, letting the smaller eddies to smear details and the larger eddies to transport the patch. Fluids 2017, 2, 17 3 of 27 In addition, there is recent, important literature on fully-non-Fickian approaches, aimed at providing prediction for the “heavy-tail” behavior of the mass transport of some substances. Despite the introduction of several recent novel approaches, the Fickian diffusion assumption based on turbulent diffusivity remains the most widely-applied framework to study the transport of a scalar in a turbulent flow. The assumption of the standard gradient diffusion hypothesis requires the estimation of the turbulent Schmidt number, Sct. This parameter is defined as the ratio of momentum diffusivity to mass diffusivity in a turbulent flow: nt−i Sct−i = (3a) Dt−i where nt-i is the eddy kinematic viscosity in the i-th direction. Hence, if the turbulent Schmidt number is known, the turbulent diffusivities can be estimated as: nt−i Dt−i = (3b) Sct−i The turbulent Schmidt number is analogous to the turbulent flow of the Schmidt number: n Sc = (4) Dm where n is the molecular kinematic viscosity of the fluid and Dm is the molecular diffusivity of the scalar within the fluid. Therefore, the turbulent Schmidt number Sct is a property of the turbulent flow, whereas the Schmidt number Sc is a property of the fluid and of the substance being diffused within the fluid. The Schmidt number Sc is usually in the order of one and of 102–102 depending on temperature for environmental flows in air and water, respectively (Table1). On the contrary, since Sct is a characteristic feature of the turbulent flow, no universal value could be established. Hence, by analogy between momentum and mass transport, Sct is often assumed as a first approximation to be equal to unity. However, empirical values different from one have been used in different studies. The SGDH is also used in Large Eddy Simulation (LES) to estimate the mass flux. Therefore, a turbulent Schmidt number for SGS motion, Sct-SGS, which is depending on the local flow characteristics,